Integral equations in the theory of levy processes

Abstract

In the article we consider the Levy processes and the corresponding semigroups. We represent the generators of these semigroups in convolution forms. Using the obtained convolution form and the theory of integral equations we investigate the properties of a wide class of Levy processes (potential, quasi-potential, the probability of the Levy process remaining within the given domain,long time behavior). We analyze in detail a number of concrete examples of the Levy processes (the stable processes, the variance damped Levy processes, the variance gamma processes, the normal Gaussian process, the Meixner process, the compound Poisson process.)